Extrapolation to product Herz spaces and some applications

This paper extends the extrapolation theory to product Herz spaces. To prove the main result, we first investigate the dual space of the product Herz space, and then show the boundedness of the strong maximal operator on product Herz spaces. By using this extrapolation theory, we establish the John–Nirenberg inequality, the characterization of little bmo, the Fefferman–Stein vector-valued inequality, the boundedness of the bi-parameter singular integral operator, the strong fractional maximal operator, and the bi-parameter fractional integral operator on product Herz spaces. We also give a new characterization of little bmo via the boundedness of the commutators of some bi-parameter operators on product Herz spaces. Even in the one-parameter setting, some of our results are new.

BEST CONSTANTS IN THE WEAK-TYPE ESTIMATES FOR UNCENTERED MAXIMAL OPERATORS

Let μ be a Borel measure on ℝ. The paper contains the proofs of the estimates and Here A is a subset of ℝ, f is a μ-locally integrable function, μ is the uncentred maximal operator with respect to μ and cp,q, and Cp,q are finite constants depending only on the parameters indicated. In the case when μ is the Lebesgue measure, the optimal choices for cp,q and Cp,q are determined. As an application, we present some related tight bounds for the strong maximal operator on ℝn with respect to a general product measure.

On Strong Maximal Operators Corresponding to Different Frames

The problem is posed and solved whether the conditions and sup θ∈[0,π/2) ∫{M 2, θ (ƒ) > 1} M 2, θ (ƒ) < ∞ are equivalent for functions (where M 2, θ denotes the strong maximal operator corresponding to the frame {OXθ, OYθ }). The results obtained represent a general solution of M. de Guzmán's problem that was previously studied by various authors.